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Byju's Answer
Standard XII
Mathematics
Nature of Roots
If a, b, c ∈ ...
Question
If
a
,
b
,
c
∈
R
+
, then both the roots of the equation
(
x
−
b
)
(
x
−
c
)
+
(
x
−
a
)
(
x
−
c
)
+
(
x
−
a
)
(
x
−
b
)
=
0
are always
A
real and positive
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B
real and negative
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C
real and equal
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D
imaginary
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Solution
The correct option is
A
real and positive
(
x
−
b
)
(
x
−
c
)
+
(
x
−
a
)
(
x
−
c
)
+
(
x
−
a
)
(
x
−
b
)
=
0
⇒
3
x
2
−
2
x
(
a
+
b
+
c
)
+
a
b
+
b
c
+
c
a
=
0
Now, for the nature of roots
D
=
4
(
a
+
b
+
c
)
2
−
12
(
a
b
+
b
c
+
c
a
)
=
4
a
2
+
4
b
2
+
4
c
2
−
4
(
a
b
+
b
c
+
c
a
)
=
2
[
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
a
−
c
)
2
]
∴
D
≥
0
⇒
Roots are real.
Let the roots be
α
,
β
α
+
β
=
2
3
(
a
+
b
+
c
)
>
0
⋯
(
1
)
α
β
=
1
3
(
a
b
+
b
c
+
c
a
)
>
0
⋯
(
2
)
From
(
1
)
and
(
2
)
,
we can conclude that roots are positive.
Suggest Corrections
0
Similar questions
Q.
If
a
,
b
,
c
∈
R
+
, then both the roots of the equation
(
x
−
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)
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+
(
x
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x
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Q.
Assertion :If equation
a
x
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+
b
x
+
c
=
0
and
x
2
−
3
x
+
4
=
0
have exactly one root common, then at least one of
a
,
b
,
c
is imaginary. Reason: If
a
,
b
,
c
are not all real, then equation
a
x
2
+
b
x
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c
=
0
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Q.
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Standard XII Mathematics
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